11630 is the First Uninteresting Number

There’s an old math paradox that says that all natural numbers are interesting, since otherwise there would have to be a smallest uninteresting number, and that in itself is pretty interesting. Of course, this is meant to show that ideas in the English language do not always translate to well-defined mathematical concepts, but let’s ignore our better mathematical sense and tinker with the idea of how interesting different numbers are a little bit. In particular, I claim that 11630 is utterly bland and uninteresting.

Why 11630?

Before saying why 11630 is uninteresting, I should probably say what I consider “interesting” to even mean. Interesting, to me, means that it has some (semi-unique) mathematical property that sets it apart from other numbers. 11 is interesting because it is prime, 16 is interesting because it is a perfect power (16 = 42), and so on. Clearly, there is some ambiguity in this definition, since one could consider composite numbers interesting, just as I considered prime numbers interesting. Additionally, do we consider 2719 interesting simply because it is prime? I’d say no, since there are hundreds of prime numbers that come before it — perhaps only the first few numbers that satisfy a given property should be considered interesting as a result of it?

Using these ideas, it seems like determining how interesting a number is would be a task perfectly suited to the Online Encyclopedia of Integer sequences (OEIS). If you’re unfamiliar with it (i.e., if you’re not a math person and have no place reading my blog), the OEIS is a database containing thousands of (you guessed it) integer sequences that have been submitted by users over the last decade or so (such as the sequence of prime numbers 2, 3, 5, 7, 11,… and the sequence of perfect powers 1, 4, 8, 9, 16,…). Presumably, if an integer is interesting then it will appear in at least one or two of the 159437 sequences contained in the database, right? Indeed, it seems that we can get a rough idea of how interesting a number is by looking at how many sequences that number appears in in the database compared to other numbers of similar size.

11630 is the first number that is not listed in a single sequence in the OEIS. It is not prime, nor is it highly composite (11630 = 2×5×1163). It doesn’t have any particularly notable residue properties, and it doesn’t come up in counting problems. It’s boring in every way, and it seems as though not a single mathematician has found a use for it in the last dozen or so years (let me know if you’ve discovered otherwise).

What Numbers are Interesting?

First off, I’m not going to deal with particularly small numbers (say in the range of 1 – 50) since, as the strong law of small numbers quips, these numbers will appear all over the place just because they’re small. You could probably argue that most (if not all) of them are interesting, so I’ll instead take a look at a couple larger numbers that are particularly interesting.

The number 421 appears in some 1894 sequences, while most numbers that size appear in about 940 sequences. This seems to indicate that 421 is a particularly interesting number, but why? What’s so special about 421? Well, it’s prime (in fact, it’s a twin prime, Pythagorean prime, cuban prime, lucky number of Euler prime, additive prime, and irregular prime), it’s congruent to 1 mod 2,3,4, 5, 6,7, 10, 12, it’s the sum of five primes, and 4212 = 4202 + 292. Similarly, 512 appears in 2116 sequences even though most numbers around 512 appear in about 800 sequences. This is perhaps less surprising than 421, since 512 = 29 = 83 = 162 + 162 is a number that somehow seems “nice” due to it being a perfect power. Additionally, 512 is a Leyland number, Harshad number, and it comes up in allsortsofcountingproblems.

What of the Paradox?

Recalling the paradox from earlier, we are now forced to ask ourselves whether or not 11630 is now interesting as a result of it being the first number not included in the OEIS. Rather than come up with an answer, I’m going to take the easy way out and let the OEIS decide. The sequence of uninteresting numbers is 11630, 12067, 12407, 12887, 13258, 13794, 13882, 13982, 14018, 14163,… Let’s submit that to the OEIS and see if they consider it to be interesting or not.

Update [June 13, 2009]: I got word back via e-mail today that this sequence didn’t make the cut. So there you have it — these numbers truly are uninteresting.

Update [November 12, 2009]: It looks like 11630 is now listed in the OEIS. Additionally, 12067 was recently added, meaning that 12407 is now the first uninteresting number.

Update [October 7, 2011]: Interested readers might want to check out this paper, which explores similar questions and mentions the numbers computed here.

Update [November 14, 2011]: The British television show QI recently aired a segment on exactly this topic. See the video here.

Update [November 22, 2013]: A whole bunch of these numbers have been added to the OEIS lately, making 14228 the new first uninteresting number.

Some of the numbers that do occur in a sequence are probably uninteresting, so we can not be sure enough that 11630 is the smallest uninteresting number to make that an interesting fact. Thanks for working this out. And the challenge is to come up with an interesting sequence that includes 11630, and many of the other small ‘uninterestings’.

I should point out that whether or not a sequence is interesting can often be independent of whether or not it contains terms that are interesting. You can have an uninteresting sequence full of interesting numbers, or an interesting sequence full of uninteresting numbers. Perhaps your sequence was rejected not because the numbers were uninteresting, but because the sequence itself was uninteresting. 😉

Was this originally your idea? If it was, it looks as if you’ve started a meme. I see reddit submissions about this literally all the time. There are two on the front page of http://www.reddit.com/r/math/ right now.

@Fletcher Tomalty – Nope, the interesting number paradox is as old as the hills, and the usual story given for its source is Ramanujan noticing that the “apparently dull” number 1729 can be written as the sum of two cubes in two different ways.

I actually noticed the two posts on reddit today (I’m a fairly active redditor) and have been fairly surprised by the amount of traffic that this post gets.

I think that a number being a sum of consecutive prime numbers is also interesting. So with your list of otherwise uninteresting numbers, we have:
12067 is the sum of all primes from 4019 up to 4027.
12407 is the sum of all primes from 113 up to 397.
13882 is the sum of all primes from 5 up to 397.
13882 is the sum of all primes from 2293 up to 2339.
13982 is the sum of all primes from 1117 up to 1213.
14018 is the sum of all primes from 3491 up to 3517.

(Also, 12887 = 112^2 + 7^3, which is kind of cool.)

So that leaves us with only 14163 from your list. I’ll admit it, that number is just plain boring

I believe you are incorrect, good sir. There CAN BE NO UNINTERESTING numbers, 11630 is interesting SINCE it IS the first number in the sequence that is uninteresting, and so on and so fourth with the rest of the numbers, any fact about a single number can be interesting, so, 11630 being the first uninteresting numbers makes it interesting.

the matter is so controversial because there is no set definition for what quallifies as interesting or uninteresting, every one is different and the way we think, we cna all have different interesting and uninteresting number IN OUR OPINION, there can’t possibly be a set uninteresting number.

I would like to nominate the number 1 as the most boring number. I arrive at this conclusion because it appears in over two thirds of the sequences. What can the other numbers discuss with the number 1? It is everywhere.

I don’t know how I happened upon this, but I did. I was looking for a cool number to use as a time of day in a digital watch in a short film I’m writing. I chose 3:14, but also considered the Golden Ratio.

Anyway, about your numbers. You chose 11630 as uninteresting, and I find something even more uninteresting at the heart of that. Literally, right in the center. The number six. 6. I think it is the first uninteresting number, and if this comment box allows enough space, I will explain why.

0 is very interesting because its zero. It makes thing real or undefinable.
1 is the first natural number. It’s number one!
2 is the first even number. It’s the first prime number and the only prime number, at that. It’s also the sign for peace.
3 is the first odd prime. Lot’s of good things come in threes; stooges, musketeers, charms. When counting large quantities of items, I find it best to run 3 at a time. 2 is too slow and four too big.
4 is the first number as can have a perfect square root. It is also useful 4 making puns. I have four eyes.
5 is five. It’s awesome in it’s own respect. It makes math easier just to see it there. All numbers associated with five include 5 or 0, and that’s it.
6 is boring, uninteresting.
7 is the number of days in a week. Seven is the most magical number of all. It inspires suspense and mysticism, and it is very melodic. “Seven”
8 is the first number with a perfect cube root. It is infinity standing proud. Eight has a lot of possibilities.
9 is cool because it is the square of 3. It’s also the “Lead note” of numbers. It leads to 10. If you multiply nine by any number, then add the resulting digits to get a single digit, it will always be 9. 9 is as perfect as it gets, and what we all strive towards. Nothing is 100%, but 99.99% is pretty good.
10 is ten. The first two digit number. The basis of the great metric system which the dumb U.S. should adopt already. Ten is the Godfather of numbers.

I’ll stop there. I believe this may prove that 6 is the ‘first’ uninteresting number. I mean, no one likes a V6 when you could have V8 power. 6 is the first failing grade on the 10 grade scale. 6 is almost in the center, but not quite. The first five birthdays are awesome, but then you turn 6 and you’re just another kid with another birthday. I could go on and on.

Let me know if you disagree. I think you’d be hard pressed to find contrary evidence. Have a good day!

6 is the only integer which can be expressed both as the multiple and as the sum of the same three integers (1, 2, 3); that seems pretty remarkable !
See also the reference to Graham’s number, the final entry in David Wells’s Penguin Dictionary of Curious and Interesting Numbers (1964).
The same book suggests that 39 is the first uninteresting number.

— the smallest absolute (weak) Euler pseudoprime (a stronger property than being a Carmichael number);
— the smallest positive integer expressible as the sum of two cubes of positive integers in two distinct ways.